Chi Peng, Shou-ceng Tian, Gen-sheng Li

State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum (Beijing), Beijing 102249, China

Abstract: To accurately determine the shedding frequency of the cavitation cloud in a submerged cavitation jet, the spectral analysis and the proper orthogonal decomposition (POD) for high-speed photography images are performed. The spectrums of 6 different kinds of image signals (the area-averaged gray level, the line-averaged gray level, the point gray level, the cavitation length, width,and area) are calculated and compared. The line-averaged gray level is found to be optimal in determining the shedding frequency but an accurate frequency can only be obtained in the stable-frequency zone where the cavitation cloud sheds. In repeated experiments, the plateau-shape distribution of the main frequency is established with a deviation of 10.8%. A revised Reynolds number Re′ is defined and the shedding frequency can be correlated to Re′ by a power law when the cavitation number is less than 0.02. This relationship is validated by the experimental data in literature. The first mode of the POD characterizes the ensemble-average of the cavitation cloud while the second mode is the major part of the cavitation cloud transient components. The modes 2-5 are organized in pairs, which confirms the periodic feature of the cavitation cloud in the submerged cavitation jet. Near the nozzle exit, the modes 2-5 are symmetrically distributed in the jet shear layer. The shedding frequency of the cloud cavitation can also be precisely determined by performing the spectral analysis of the weighting coefficients of the mode 2. This paper shows that the two parameters, namely, the line-averaged gray level and the weighting coefficients of the mode 2, can be confidently used to calculate the shedding frequency of the cavitation cloud in a submerged cavitation jet.

Key words: Cavitation jet, shedding frequency, high-speed photography, spectral analysis, proper orthogonal decomposition (POD)

Introduction

The cavitation happens when the vapor cavities,called the cavitation bubbles, are formed in the liquid as the local pressure drops below the saturated vapor pressure[1]. Under a high ambient pressure, the cavitation bubble will collapse, generating an extremely high pressure and temperature (up to several GPa and thousands of K)[2]. If the bubble collapses in the places close to a solid boundary, high-speed microjets and shock waves will be produced, which can lead to permanent deformations, pits, and craters[3]. The cavitation can cause noise, vibration, low efficiency,and erosion in the equipment, such as pump, turbine,rudder, valve, and nozzle.

Although the cavitation is often not a desirable phenomenon in the hydraulic engineering, the submerged cavitation jet (SCJ) may take advantage of the energy of the cavitation bubble collapse, with extensive applications and researches since the 1980s.The cavitation is induced in the nozzle throat of the SCJ where the local pressure drops dramatically due to the Bernoulli's Principle. The microscopic roughness of the inner surface of the nozzle, the small solid impurities, and the dissolved gas can provide the sites for the heterogeneous nucleation of the bubbles[4].The cavitation bubbles grow and reside in the jet shear layer, forming the large-scale coalesce, called the cavitation cloud. As the cavitation bubbles approach the target, they will collapse at the target surface due to the stagnation pressure of the jet. It has been shown that the SCJ can improve the drilling rate for the petroleum wells[5], perform shotless peening, dispose of organic waste water[6], and clean underwater pipelines[4], etc..

There are principally two ways to investigate the SCJ. One is the erosion test that focuses on the mass loss, the damage, and the failure patterns of various materials. Related parameters include the nozzle geometry, the upstream pressure, the standoff distance,the cavitation number, and the temperature, and their influences on the jet erosion ability were investigated[7-10]. Another method is the visualization of the cavitation cloud in the SCJ.

With the development of the visualization techniques, the dynamic photos with a high temporal resolution are obtained. In recent years, the high-speed photography becomes a popular visualization method in the SCJ studies. It was firstly adopted by Chahine et al.[11], who found that there was periodic shedding of the cavitation cloud in the SCJ. This periodic behavior was verified by many studies[12-16]. A typical cycle of the cavitation cloud contains the inception, the development, the shedding and the collapse. The hydraulic conditions such as the upstream pressure and the cavitation number would affect the size and the shedding frequency of the cavitation cloud.Soyama et al.[12]observed that the shedding frequency of the cavitation cloud decreases, while the maximum length increases, with the increase of the upstream pressure. The unsteady behavior of the cavitation cloud was thought to be closely related to the pressure gradient in the shear layer. Hutli and Nedeljkovic[13]calculated the shedding frequency under different upstream pressures and with different nozzle geometries. They confirmed the inverse relation between the upstream pressure and the shedding frequency, obtained by Soyama et al.[12]. Nishimura et al.[14]established a similarity law between the shedding frequency and the hydraulic parameters. The shedding frequency was found to be inversely proportional to the characteristic width of the cavitation cloud and directly proportional to the square of the upstream pressure. The vortex shedding was proposed as a cause of the periodic shedding of the cavitation cloud. Wright et al.[15]investigated the size, the width, the length, the shedding frequency,and the front velocity of the cavitation cloud at different Reynolds numbers. They produced the SCJ by ejecting a small amount of water through a nozzle with the help of the high-pressure nitrogen, thus avoiding the fluctuation of the upstream pressure.With a plunger pump to provide the upstream pressure,the low-frequency signals (usually below 100 Hz)induced by the pump pressure fluctuation can be obtained. Sato et al.[17-18]utilized the frame difference method to analyze the shedding, transition, and collapse processes of the cavitation cloud with a horn-type nozzle. They suggested that the re-entrant jet formed by the downstream cavitation cloud collapse was responsible for the cavitation cloud shedding. The re-entrant jet periodically “pinched off”the cavitation cloud at the nozzle exit[19], very similar to the cavitation cloud shedding on hydrofoils[20].

The shedding frequency of the cavitation cloud is an important parameter to characterize the periodic behavior of the SCJ. To determine the shedding frequency from high-speed images, several kinds of the image signals were used in previous studies: (1)the area-averaged gray level in a Region of Interest(ROI)[21], (2) the line-averaged gray level along a cross line of the cavitation cloud[15,20,22], (3) the point gray level at the jet centerline[14,19], (4) the cavitation cloud length[16-17], and (e) the cavitation cloud width[13,16],as well as, (f) the area of the cavitation cloud[15],which is also considered in this paper. Usually, the main frequency (the characteristic frequency) of the cavitation cloud shedding is acquired by performing the fast Fourier transform (FFT) of the time series signals of the images. However, it remains to be investigated whether the shedding frequencies obtained from different kinds of image signals are consistent . Besides, previous studies did show that the main frequencies in different flow regions had obvious deviations[14,16]. For the SCJ, theoretically speaking, there should only be one frequency that characterizes the shedding of the cavitation cloud.

The proper orthogonal decomposition (POD) is an efficient method to identify the spatial and temporal coherent structures of the flow fields. With the method, a time series of the vector or scalar field is decomposed into a set of basic spatial modes and temporal coefficients[23]. Large scale structures and main patterns of the field can be highlighted in the POD modes. Besides, the coefficients contain the temporal information about the whole flow process.Therefore, the POD enjoys widespread applications in the fluid mechanics[24-25]and recently it was introduced to the study of the cavitation flow. Danlos et al.[22]proposed to use the coefficient of the second POD mode to distinguish different cavitation regimes in a convergent-divergent nozzle. Prothin et al.[20]conducted the POD and the dynamic mode decompositions analysis of the cavitation flows over hydrofoils and emphasized the spatio-temporal features of the cavitation cloud. Watanabe et al.[26]calculated the spatial modes of the cavitation cloud in the SCJ and found that the cavitation cloud symmetrically resided in the jet shear layer. The weighting coefficients contain the temporal information about the SCJ, which can be used to explore the frequency feature of the transient cavitation flow[22].

In this paper, we are going to compare the cavitation cloud shedding frequencies calculated from different kinds of image signals to reveal the requirements that are needed to obtain the real shedding frequency. Conventional kinds of signals directly obtained from the high-speed images and one novel kind of signals (the weighting coefficients of the mode 2) from the POD analysis are tested. What is more, we try to find the relation between the shedding frequency and the hydraulic condition based on the data obtained by previous studies.

1. Experimental setup

The test rig is schematically shown in Fig. 1. The upstream pressure is provided by a high-pressure plunger pump (KMT Streamline SL-V50), with an inbuilt bladder accumulators to suppress the pressure fluctuation. The maximum flow rate and pressure are 3.79 L/min and 380 MPa, respectively. The upstream pressure is measured at the outlet of the plunger pump by a pressure gauge (with an accuracy of ±0.3% FS).The pressure loss from the pump to the nozzle is not considered. A turbine flowmeter (with an accuracy of±0.5% FS) is used to measure the flow rate. A cylindrical nozzle with a dismountable mixture chamber and an extension sheath is used, with nozzle throat diameter of 0.3 mm. The abrasive feeding hole is sealed with a plug. Surprisingly, it was found that,with the mixture chamber and the extension sheath,the cavitation became more intensive, probably due to the pressure pulsation and the wave resonance as in the self-resonance cavitation nozzle[7,8,18]. A cylindrical water vessel made of transparent acrylic resin(polymethyl methacrylate), with diameter of 300 mm and height of 800 mm, is used to achieve the submerged condition. The tap water is used, with the water temperature of 16°C-18°C during the experiments, measured by an electronic thermometer (with a resolution of 0.5°C, and the maximum error of 1°C).

Fig. 1 (Color online) Test rig (mm)

A high-speed camera (Phantom V310, lens:AF80-200 mmF/2.8D) is used to visualize the cavitation cloud, with a frame rate of 10,000 fps and the image size of 128×208 pixels corresponding to a real area of 41.0 mm×66.6 mm (1 pixel≈0.32 mm). The illumination is provided by a 1 000 W halogen lamp placed right above the camera. As a result, the cavitation cloud appears as the bright region in the images. The lateral and axial coordinates x and y are defined as in Fig. 1, with the origin at the nozzle exit. The standoff distance is equal to y and the nondimensional standoff distance (NSD) is y/throat diameter.

To verify the repeatability of the results, 10 experiments are conducted under the same hydraulic condition. The upstream pressure is fixed at 60±0.15 MPa. The flow rate is 1.21±0.06 L/min. The nozzle discharge coefficient is 0.85± 0.04. The water in the supply tank is prepared a day before the experiment to be exposed to the atmosphere for 24 h.As the cavitation bubbles collapse, there would be more cavitation nuclei after each experiment. To keep a constant water quality, the water in the vessel is replaced by the fresh tank water before the next test.The high-speed filming would not begin until the upstream pressure is maintained at the desired value.

2. Data processing method

2.1 Image process

In order to avoid the possible influence from the non-uniform illumination and the lens blur, the data of all images are subtracted by the data of a background image, which is taken before the SCJ is initiated. The irrelevant parts in the images are removed and the final size of each image is 58×204 pixels (18.6 mm×65.3 mm). Then they are transformed into double-type gray-scale images. Each pixel has a gray level between 0 (white) and 1 (black). Gray images are further turned into binary images to calculate the length, the width, and the area of the cavitation cloud.

2.2 High-speed image signals

1 000 successive images are picked out from each experiment. Six kinds of image signals for performing the FFT are shown in Fig. 2. (1) The area-averaged gray level of the ROI: the average gray level of all the pixels in a ROI. The whole flow field can be divided into several ROIs (for example, 6 ROIs in Fig. 2(a)), and each ROI has an averaged gray level.Different divisions of ROIs in y direction are tested.(2) The line-averaged gray level: the gray level averaged from the pixels along a line parallel to x axis, e.g. y = 68 pixel (22.8 mm) in Fig. 2(a). (3)The point gray level: the gray level of a point at y axis. (4) The length of the cavitation cloud. (5) The width of the cavitation cloud. (6) The area of the cavitation cloud.

Fig. 2 (Color online) High-speed image signals

2.3 Principle of POD

The POD is essentially a linear approximation method to disintegrate the transient vector field (e.g.,the velocity field) or the scalar field into the sum of a series of orthonormal basis functions (the modesmφ),and corresponding weighting coefficients. The modes contain the spatial information while the weighting coefficients contain the temporal information about the field. The field at a specific time is called a snapshot’ and the number of modes is equal to the number of snapshots. In this paper, the number of snapshots is 1 000, which is much smaller than the number of spatial points (11 832 pixels). Thus, the snapshot method is used in this paper.

Each gray image of the transient cavitation cloud can be regarded as a snapshot (a gray level field) G.The principle is to decompose K snapshots(k)G into a linear combination of M modesmφ and K×M weighting coefficients

where i and j indicate the position of the pixel, k means kth snapshot, I=204, J=54.

Then the spatial correlation matrix W is

To find a sequence of orthonormal modes that reflect the basic structures of the field, we let the following summation take a minimum value

(2) Project U onto each eigenvector βm(m =1,2,…,M). The modes are the normalized projections.

The modes represent the basic compositions of the flow field. The snapshots can be reconstructed through Eq. (1), i.e., summing the products of every mode with the corresponding coefficients. Here, the modes of the gray level field illustrate the fundamental compositions of the cavitation cloud. In particular, if the flow process is dominated by a periodic behavior, the low-order modes are organized in pairs, which represent the same orthogonal component of the periodic phenomenon.

The weighting coefficients are computed by projecting the original snapshots onto the modes,which are assembled in a matrix as

which reflects the contribution of each mode to the reconstruction of all snapshots, i.e., the relative importance in all modes. The modes are arranged in the order of decreasing energy fraction. A decomposition is considered to be converged if an increase in the number of snapshots does not lead to significant changes of the energy fractions of the low-order modes. Here, the energy fractions of the modes represent their contributions to the reconstruction of the cavitation cloud snapshots. The low-order modes are the basic and dominant compositions of the dynamic cavitation cloud.

The modes and the coefficients are related to the ensemble-averaged field through

The relevance index Rican be used to evaluate the similarity between two modes, two snapshots or one mode and one snapshot. It is defined as[23]

where () means inner product of two matrixes. Riis between -1 and 1. The closer theis to 1, the more similar the two matrixes are to each other.

3. Results and analyses

3.1 Periodic behavior of cavitation cloud in SCJ

Figure 3 shows a typical periodic behavior of the cavitation cloud in the SCJ. In Fig. 3(b), the gray level is proportional to the local vapor fraction. The shedding of the former cavitation cloud and the initiation of a new cavitation cloud take place in a time between 0 ms and 0.1 ms. The expanding and moving downstream of the new cavitation cloud takes a time from 0.1 ms to 0.7 ms. It can be seen that the vapor fraction in the cavitation cloud gradually decreases as the cavitation cloud moves downstream.The “re-entrant motion” (the shrinking of the upper part of the cavitation cloud before shedding) is displayed in dashed square A, which was also captured by Hutli et al.[13]and Sato et al.[17]. At the same time, the shedding process of the cavitation cloud is slowed down by the retardation effect of the water. It shrinks as a large number of bubbles collapse.The shed cavitation cloud finally disappears in the downstream region (detailed in B). Between 0.7 ms and 0.8 ms, the shedding and the initiation of the next cavitation cloud start again. The image of 0.7 ms roughly coprresonds to that of 0 ms, both at the same phase of a period. The bump-shape protrusions shown in dotted squares C1and C2are the projections of three dimensional cavitation structures, which are closely related to the coherent structures in the shear layer.

Fig. 3 (Color online) Periodic behavior of cavitation cloud

3.2 Spectral analysis of high-speed image signals

As is mentioned in 2.2, the whole flow field can be divided into different numbers of the ROIs, each with a specific length in y direction. It is divided into 1 ROI in Fig. 4(a), 6 ROIs in Fig. 4(b), 17 ROIs in Fig.4(c), and 68 ROIs in Fig. 4(d). Usually, different ROIs have disparate power spectrum densitys (PSDs).However, some general trend can be discovered from Fig. 4. When the ROI is large (Figs. 4(a), 4(b)), the spectral energy is more concentrated in the low-frequency domain, with the high-frequency noise of the signals suppressed. It is hard to determine the main frequency in Fig. 4(a), where are multiple peaks with comparable magnitudes. As the ROI gets smaller,the spectral energy gradually transfers to the high-frequency domain and finally converges at the characteristic frequency (1 350 Hz) (Figs. 4(c), 4(d)and 6(b)). Note that the line-averaged gray level is the same when the ROI length=1 pixel. This indicates that though the axial (y-direction) average of the gray level signal can filter the high-frequency noise[21], it amplifies the low-frequency noise and can lead to a great deviation in the main frequency. The large area of the ROIs is a possible reason for the low-frequency peaks obtained in Ref. [21].

Fig. 4 (Color online) PSDs of area-averaged gray level with different ROI lengths

The main frequency distributions derived from the ROIs of different lengths are shown in Fig. 5. The position of each ROI y is obtained as its geometry center. For comparison, all ROIs in Fig. 5 include y=21.8 mm (68 pixel, NSD 72.5) line. Apparently,the main frequency varies with the standoff distance[14].However, all curves have a similar “plateau” shape.The main frequency is only 10 Hz if y is less than 8.6 mm (NSD 28.8 ) or large r th an 4 0.3 mm ( NSD 134.4).When y is between 8.6 mm and 40.3mm,the main frequency fluctuates and it is impossible to find a convincing main frequency that can represent the cavitation cloud shedding. This plateau-shape distribution also applies to the PSDs of other image signals (Fig. 9).

Fig. 5 (Color online) Main frequency distributions of areaaveraged gray level with different ROI lengths

Fig. 6 (Color online) Line-averaged gray level at y = 21.8 mm and its PSD

Fig. 7 (Color online) Line-averaged gray level and its PSD obtained by Wright et al.[15]

Fig. 8 (Color online) PSDs of different image signals

Fig. 9 (Color online) Main frequency distributions derived from line-averaged gray level, point gray level, and cavitation cloud width

The line-averaged gray level signal at y=21.8 mm and its PSD are given in Fig. 6. The main frequency is 1 350 Hz and the spectral energy converges around the main frequency. No obvious peaks can be seen in the low-frequency domain(below 100 Hz). Figure 7 show the experimental results obtained by Wright et al.[15], who used the high-pressure nitrogen to accelerate the SCJ.Compared with Fig. 6(b), clear low-frequency peaks can be observed in Fig. 7(b). This means that the low-frequency noise could not result from the pump pressure fluctuation[16]. A possible explanation is that the position where the line-averaged gray level is obtained is too close to the nozzle (Fig. 6 in Ref. [15]),possibly within the transient-frequency zone B of Fig.9. In fact, Fig. 7(b) is very similar to Fig. 10(b).

The PSDs of the point gray level, the cavitation length, width, and area are shown in Fig. 8. In general,Figs. 6(b), 8(a) and 8(c) have a similar “mount” shape and a same main frequency of 1350 Hz, indicating the equivalency of the three kinds of signals (the line-averaged gray level, the point gray level, and the cavitation cloud width). This can be further manifested in Fig. 9. On the other hand, as compared with Figs. 8(a), 8(c) and 6(b) shows that more spectral energy is concentrated around the main frequency,showing the advantage of performing the radial (xdirection) average of the gray level signal. Although the PSD of the cavitation cloud length (Fig. 8(b)) also has a mount shape, its spectral energy is more scattered and it is hard to decide the main frequency from two frequencies with comparable magnitudes(940 Hz, 1 100 Hz). Figure 8(d) is nearly identical to Fig. 4(a), which reveals the equivalence between the cloud area and the globally averaged gray level.Similarly, several peaks with comparable magnitudes exist, making it impossible to determine the shedding frequency from Fig. 8(d). Therefore, neither the cavitation cloud length nor its area is an eligible kind of signals to determine the accurate shedding frequency. Among the other four kinds of image signals, the line-averaged gray level can be considered as the best option based on the following three reasons:(1) the convergence of the spectral energy at the characteristic frequency with deceasing ROI length for the area-averaged gray level, (2) the spectral energy of the line-averaged gray level is more easily converged as compared with the point gray level and the cavitation cloud width, (3) in the distribution curves (Fig. 9), there are occasionally abrupt protrusions for the point gray level and the cavitation cloud width (such as the sudden drop of the main frequency near y=13.6 mm for the point gray level signal). In contrast, the line-averaged gray level distribution has a more regular shape, and there is a zone(y=17.6 mm-28.2 mm)with a very stable main frequency of 1 350 Hz.

According to Fig. 9, the whole flow field can be divided into 5 zones in y direction, labeled as A to E.The main frequencies in A (0 mm-8.6 mm, NSD 0-28.8) and E (40.3 mm-64.6 mm, NSD 124-216) are very small, indicating the dominance of the DC signal.Zones A and E are called the “0-frequency” zone. The main frequency remains at 1350 Hz in zone C(17.6 mm-28.2 mm, NSD 58.7-93.9), which is the“stable-frequency” zone. Between the 0-frequency zone and the stable-frequency zone, there are two transient zones B and D where the main frequency fluctuates. The typical PSDs of A, B, D and E are given in Fig. 10. In zone A, near the nozzle, the cavitation cloud is very compact (Fig. 3(b)) and its periodic behavior is less evident. Zone C is found to be the location of the cavitation cloud shedding, while zone E corresponds the location where the cavitation cloud collapses. The zones C and E were also captured in Ref. [14].

The repeatability of the main frequency distribution is tested. It can be observed in Fig. 11 that all distribution curves have a “plateau shape” composed of the 0-frequency zones A and E, the stablefrequency zone C, and the transient zones B and D.Sometimes, the transition zone B might be missing.The persistent existence of the stable-frequency zone and that it corresponds to the locations of the cavitation cloud shedding confirm that only the main frequency obtained in the stable-frequency zone is the real shedding frequency of the cavitation cloud. The

Fig. 10 (Color online) PSDs of line-averaged gray levels

Fig. 11 (Color online) Main frequency distributions of lineaveraged gray level from repeated experiments

locations of the zones A-E have some shifts in the experiments, and the intersection of the stablefrequency zone is found to be 17.9 mm-24.3 mm(NSD 59.7-81.1) from the nozzle exit. The length of the stable-frequency zone is 10.8±1.8 mm (the non-dimensional length 35.8±6.1). Although the hydraulic conditions are kept the same in the repeated experiments, the shedding frequency ranges from 1 130 Hz to 1 590 Hz. The average value is 1 340 Hz with the standard deviation of 10.8%. Some factors will affect the cavitation intensity and the periodic behavior of the cavitation cloud in the SCJ, including the inner surface roughness of the nozzle, the residual gas bubbles in the vessel, the impurities and the dissolved gases in the incoming water jet.Nevertheless, a standard deviation of 10.8% is considered acceptable according to the rule of thumbs:the cavitation measurements with a standard deviation within 15% is regarded as satisfactory.

3.3 The relationship between shedding frequency and hydraulic conditions

Experimental data of the cavitation cloud shedding frequency in the SCJ are taken from the literature. The experiments were conducted under various hydraulic conditions, with the nozzle diameter ranging from 0.3 mm to 4.0 mm, the upstream pressure of 0.200 MPa-70.000 Mpa, the ambient pressure of 0.101 MPa-1.000 Mpa and the temperature of 288 K-293 K. To take account of the effects of the nozzle size and the ambient pressure, a revised Reynolds number Re′ is defined

where prefis the reference pressure (the atmospheric pressure), pambat the given temperature. In cases when the fluid temperature is not available, it is assumed to be 293 K. If the nozzle throat velocity is not given, it is calculated based on the pressure difference. The discharge coefficient is 0.64 for the cylindrical nozzle[12]and 0.9 for the conical and horn nozzle (the typical discharge coefficients of the conical nozzles range from 0.85 to 0.95, so we think 0.9 is a fair hit). The cavitation number

A power-law relationship between the shedding frequency and Re' is established as shown in Fig. 12

Although all data of σ were acquired in different test rigs, those less than 0.02 are found to perfectly obey Eq. (11), except for the three cases of σ larger than 0.2. Most of the data are obtained using the point gray level[14]and the line-averaged gray level[15],while the use of other variables would lead to some deviation. Note that in cases of the erosion experiment,it is found that the frequency of the cavitation cloud impact (shedding) increases with the increase of the jet velocity[1], which is contrary to the trend of Fig. 12,due to the boundary effect of the erosion sample.

Fig. 12 (Color online) Shedding frequency vs. Re′

3.4 POD analysis of cavitation cloud snapshots

A convergence test is carried out to determine the number of snapshots that is necessary to validate the decomposition. As shown in Fig. 13, the energy fractions of the modes 1-5 gradually converge with the increasing number of snapshots, and the relevance indexes turn into unity, meaning that the further increase of snapshots will not lead to a noticeable difference of the modes. Therefore it is concluded that under the present experimental condition, 1000 snapshots are enough to achieve the convergence for all the datasets. 93.5% of the gray level energy is in the first five modes, which represent the fundamental compositions of the cavitation cloud gray level field.The energy fraction of the mode 1 is 0.86. According to Eq. (8), the mode 1 should be a good approximation of the ensemble-averaged gray level field. In fact, Ribetween the mode 1 and the ensemble-averaged gray level field is 0.9999 (Fig. 14). This demonstrates the validity of the present POD results. It is noted that the mode 1 resembles a typical potential core in the water jet, suggesting a close relation between the cavitation cloud and the high-speed potential core. The modes 2-5 shown in Fig. 14 are organized in pairs,confirming the periodic feature of the cavitation cloud in the SCJ. The mode 2 comprises two counterrotating lobe-structures that are characteristic of the spatial periodic shedding. The mode 3 represents the same structures with a delay (a spatial shift) in the downstream direction, which is a classical feature of the cavitation shedding[20]. The higher order modes also follow this rule. Generally speaking, when y＜40.0 mm (133.3 NSD), the modes are symmetrically distributed in the shear layer of the jet, as is consistent

Fig. 13 (Color online) Convergence test of POD

Fig. 14 (Color online) Average gray field and modes 1-5

Fig. 15 (Color online) PSDs of the weighting coefficients of modes 1-4

with the previous findings[9,26].The expansion and the shrink of the cavitation cloud are governed by the turbulent pressure fluctuation in the shear layer. At downstream (y＞40.0 mm), the shear layer pressure fluctuation becomes weak while the ambient pressure dominates the collapse process of the cavitation cloud,so the symmetrical distribution disappears.

In a velocity field, the higher modes (other than mode 1) correspond to the turbulent components of the flow field while the mode 2, with the most turbulent energy, is the major part of the turbulent components[24]. Analogically, in the present gray level field, the modes higher than mode 1 reflect the transient components of the cavitation cloud and the mode 2 is the major part of the transient components.The gray level distribution in the mode 2 can be regarded as the main patterns of the changing part of the cavitation cloud. Figure 14 shows the positions of the five frequency zones A-E in the mode 2 and it is found that they are related to that mode. No obvious cavitation cloud change is observed in the 0-frequency zone A. Consequently, the main frequency in A is nearly 0. As for the 0-frequency zone E, it starts from the location of the cavitation cloud collapse. In the stable-frequency zone C, not only the effect of the shear layer pressure fluctuation is prominent (the mode is distributed at the side shear layers), but also most of the cavitation cloud shedding occurs in this zone.

Based on the analysis of 3.2, it is clear that to determine an accurate shedding frequency, both the right kind of the image signal (the line-averaged gray level) and the right position (within the stablefrequency zone) are necessary. To leave out the trouble of locating the stable-frequency zone, we propose to use the weighting coefficients of the mode 2 to calculate the shedding frequency. The PSDs of the weighting coefficients of modes 1-4 are depicted in Fig.15. As shown in Fig. 15(b), the PSD of the mode 2 coefficients is very similar to that in Fig. 6(b)and the main frequencies are the same[22]. This finding holds true for all repeated experiments. In comparison,there are still some low-frequency peaks in Fig. 15(b),but the spectral energy in the high-frequency domain is much less, because the high-frequency noises are filtered into the high-order modes. Moreover, it is interesting to find that the PSD of the mode 1 coefficients are nearly the same as the PSDs of the area-averaged gray level ((Fig. 4(a)) and the cavitation cloud area (Fig. 8(d)). That is to say, the global signals cannot represent the substantial periodic nature of the cavitation cloud. In contrast, they represent the characteristics of the ensemble-average flow field. As mentioned in 3.2, the spatial averaging in the flow direction can filter out the high-frequency signals[21],but at the same time some low-frequency signals might be introduced, that belong to the ensembleaverage flow field. This can disturb the determination of the shedding frequency. The main frequency derived from the large ROI is not the real shedding frequency (Fig. 4(b)). In this way, choosing a proper ROI size should be an essential issue in the cavitation flow studies. The main frequencies of the mode 3 and the related coefficients are the frequencies of some minor periodic phenomena and can sometimes be detected in the transient-frequency zone as shown in Fig. 10(b).

4. Conclusions

The cavitation cloud in the SCJ has a periodic behavior, which is visualized by the high-speed photography. The spectral analysis and the POD method can be used for a more accurate determination of the cavitation cloud shedding frequency.

The spectrums and the main frequencies of 6 kinds of image signals (the area-averaged gray level,the line-averaged gray level, the point gray level, the cavitation length, width, and area) are compared. It is found that the axial (flow-direction) averaging of the gray level signal can introduce some low-frequency noise in the PSD and lead to errors in the determination of the shedding frequency, which could possibly be a reason for some of the low-frequency peaks obtained by De Giorgi et al.[21]. The point gray level, the cavitation cloud width, and the lineaveraged gray level share some equivalence while the line-averaged gray level is optimal in determining the accurate shedding frequency. However, an accurate shedding frequency can only be obtained in a stablefrequency zone. In the repeated experiments, the existence of the stable-frequency zone is con- firmed.The shedding frequency under the present hydraulic condition is 1 350 Hz±10.8%. The position of the stable-frequency zone shall change with the hydraulic conditions. A revised Reynolds number Re′ that takes account of the ambient pressure is defined and the shedding frequencies obtained in literature can be satisfactorily correlated to Re′ by a power law when the cavitation number is less than 0.02.

The POD analysis can highlight the spatialtemporal characteristics of the cavitation cloud in the SCJ. The mode 1 characterizes the ensemble-average of the cavitation cloud gray field while the higher modes show the transient components. The mode 2 is the major part of the cavitation cloud transient component. The modes 2-5 are organized in pairs,which confirms the periodic feature of the cavitation cloud. Near the nozzle exit, the modes 2-5 are symmetrically distributed in the shear layer of the jet.The stable-frequency zone is where the cavitation cloud sheds. The characteristic shedding frequency derived from the mode 2 coefficient is equal to the frequency derived from the line-averaged gray level,as is consistent with the study of Danlos et al.[22]. The POD mode 2 coefficient and the line-averaged gray level are two convenient and reliable variables to calculate the shedding frequency of the cavitation cloud in the SCJ.